Mafunso ndi Zitsanzo za Ma Circles ndi Tangents
Mabwalo ndi ma tangent ndi mitu iwiri yomwe imakambidwa kawirikawiri mu masamu, makamaka pasukulu ya sekondale. Kumvetsetsa lingaliro ndi kugwiritsa ntchito ma tangent m'mabwalo ndikofunikira kwambiri kuti muwonjezere chidziwitso chanu cha geometry. Nkhaniyi ipereka zitsanzo za mavuto ndi zokambirana pa mabwalo ndi ma tangent kuti owerenga amvetsetse bwino.
Chiyambi cha Chiphunzitso cha Ma Circles ndi Tangents
Mzere wozungulira
Bwalo ndi gulu la mfundo zomwe zili mu ndege zomwe zili kutali ndi mfundo yokhazikika yotchedwa pakati pa bwalo. Mtunda wokhazikika uwu umadziwika kuti radius ya bwalo. Mwa masamu, bwalo lingathe kufotokozedwa ndi equation:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
kumene \((a, b)\) ndi ma coordinates a pakati pa bwalo ndipo \(r\) ndi radius.
Tangent
Mzere wozungulira ndi mzere womwe umakhudza bwalo pamalo amodzi. Malo awa amatchedwa malo ozungulira. Khalidwe lalikulu la tangent ndilakuti limakhala lolunjika ku utali wozungulira womwe umatengedwa kuchokera pakati pa bwalo mpaka pamalo ozungulira.
Mafunso ndi Kukambirana Zitsanzo
Funso 1: Kudziwa Equation ya Mzere wa Tangent
Funso:
Yapatsidwa bwalo lokhala ndi pakati pa \( (2, 3) \) ndi radius 5. Dziwani equation ya mzere wa tangent ku bwalo lomwe lili pamalo \( P \) ndi ma coordinates \( (5, 7) \).
Kukambirana:
Gawo 1: Onetsetsani kuti mfundo \( P \) ili pa bwalo.
Kuti muwone ngati \( P (5, 7) \) ili pa bwalo lokhala ndi pakati \( (2, 3) \) ndi radius \( 5 \), sinthani ma coordinates a \( P \) mu equation ya bwalo:
\[ (x – 2)^2 + (y – 3)^2 = 5^2 \]
\[ (5 – 2)^2 + (7 – 3)^2 = 25 \]
\[ 3^2 + 4^2 = 25 \]
\[ 9 + 16 = 25 \]
Popeza kufanana kuli koona, mfundo \( P \) ili pa bwalo.
Gawo 2: Dziwani momwe radius imadutsa mu \( (2, 3) \) ndi \( (5, 7) \):
\[ m_{radius} = \frac{y_2 – y_1}{x_2 – x_1} = \frac{7 – 3}{5 – 2} = \frac{4}{3} \]
Gawo 3: Kupindika kwa mzere wozungulira womwe uli wolunjika ku kupindika kwa radius (kupindika kwa chinthucho ndi -1):
\[ m_{tangent} = -\frac{1}{m_{radius}} = -\frac{1}{\frac{4}{3}} = -\frac{3}{4} \]
Gawo 4: Dziwani equation ya mzere wa tangent pogwiritsa ntchito mfundo \( P (5, 7) \):
\[ y – y_1 = m (x – x_1) \]
\[ y – 7 = -\frac{3}{4} (x – 5) \]
Chepetsani:
\[ y – 7 = -\frac{3}{4}x + \frac{15}{4} \]
\[ 4y – 28 = -3x + 15 \]
\[ 3x + 4y – 43 = 0 \]
Kotero, equation ya mzere wa tangent ndi:
\[ 3x + 4y – 43 = 0 \]
Funso 2: Kudziwa Mfundo ya Tangency kuchokera ku Line Equation
Funso:
Kupatsidwa bwalo lokhala ndi equation \( x^2 + y^2 = 25 \) ndi mzere \( y = \frac{3}{4}x + 2 \). Dziwani malo olumikizirana pakati pa mzere ndi bwalo.
Kukambirana:
Gawo 1: Sinthani equation ya mzere kukhala equation ya bwalo:
Chiyerekezo cha bwalo:
\[ x^2 + y^2 = 25 \]
Lowetsani \( y = \frac{3}{4}x + 2 \) mu equation yozungulira:
\[ x^2 + \left(\frac{3}{4}x + 2\right)^2 = 25 \]
\[ x^2 + \left(\frac{9}{16}x^2 + \frac{12}{4}x + 4 \right) = 25 \]
\[ x^2 + \frac{9}{16}x^2 + \frac{6}{2}x + 4 = 25 \]
\[ x^2 + \frac{9}{16}x^2 + 3x + 4 = 25 \]
Gawo 2: Fewetsani equation:
\[ 16x^2 + 9x^2 + 48x + 64 = 400 \]
\[ 25x^2 + 48x + 64 – 400 = 0 \]
\[ 25x^2 + 48x – 336 = 0 \]
Gawo 3: Kupeza mizu pogwiritsa ntchito njira ya quadratic:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
\[ a = 25, b = 48, c = -336 \]
\[ x = \frac{-48 \pm \sqrt{48^2 – 4 \cdot 25 \cdot (-336)}}{2 \cdot 25} \]
\[ x = \frac{-48 \pm \sqrt{2304 + 33600}}{50} \]
\[ x = \frac{-48 \pm \sqrt{35904}}{50} \]
\[ x = \frac{-48 \pm 189.501}{50} \]
Kusankha \( x \) kovomerezeka kutengera mfundo yolunjika (imodzi yokha \( x \) ipanga mfundo yolunjika):
\[ x = \frac{141.501}{50} \pafupifupi 2.83 \]
\[x \pafupifupi 2.83 \]
Gawo 4: Sinthani \( x \) mu equation ya mzere kuti mupeze \( y \):
\[ y = \frac{3}{4}(2.83) + 2 \]
\[ y \pafupifupi 2.12 + 2 \]
\[ y \pafupifupi 4.12 \]
Kotero, mfundo yolumikizana pakati pa mzere \( y = \frac{3}{4}x + 2 \) ndi bwalo \( x^2 + y^2 = 25 \) ndi \( (2.83, 4.12) \).
Mapeto
Kudziwa bwino mfundo za mabwalo ndi ma tangent kumaphatikizapo kumvetsetsa maziko a geometry ndi luso lotha kuthetsa mavuto pogwiritsa ntchito ma equation a masamu. Mavuto ngati omwe ali pamwambapa amathandiza ophunzira kuchita masewera olimbitsa thupi pogwiritsa ntchito chiphunzitso m'mikhalidwe yeniyeni. Ndi machitidwe okhazikika, ophunzira akuyembekezeka kumvetsetsa ndikuthetsa mavuto mosavuta.
